Interesting thing I've noted

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In summary, the conversation discusses the occurrence of powers of 2 being almost in the middle of consecutive squares and asks about the next occurrence. It is mentioned that a quick computer search reveals no such cases with exponents less than 1000 and suggests looking for other patterns. The conversation also briefly mentions the relation between sums and differences of two powers and shares a link to the OEIS sequence A215797. The original post violated forum rules and was deleted.
  • #1
karpmage
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2^2-1 = 2^2-1 = 1^2+2
2^3-1 = 3^2-2 = 2^2+3
2^5-1 = 6^2-5 = 5^2+6
2^13-1 = 91^2-90 = 90^2+91

I realize that x^2-(x-1)==(x-1)^2+(x)

2,3,5 and 13 are all the powers of mersenne primes, and are Fibonacci numbers as well.
It'd be interesting to see what's the next power of 2 that satisfies this equation.
 
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  • #2
Hi there, you have observed some cases of powers of 2 being almost in the middle of consecutive squares, and you ask about the next occurence. A quick computer search reveals no such cases with exponent less than 1000. So unless either I or my computer made a mistake, which may very well happen, you may want to look for other patterns instead.

What happened to your post about sums and differences of two powers? I had written a reply, including this http://oeis.org/A074981http://, and the relation 30=832-193, and some more, and then I couldn't post it.
 
  • #3
Sorry, I posted this to another maths forum as well, and it turns out there are no other numbers. Here's the link I was sent:
http://oeis.org/A215797
 
  • #4
Norwegian said:
Hi there, you have observed some cases of powers of 2 being almost in the middle of consecutive squares, and you ask about the next occurence. A quick computer search reveals no such cases with exponent less than 1000. So unless either I or my computer made a mistake, which may very well happen, you may want to look for other patterns instead.

What happened to your post about sums and differences of two powers? I had written a reply, including this http://oeis.org/A074981http://, and the relation 30=832-193, and some more, and then I couldn't post it.

My last post violated forum rules, I am going to repost a revised version of it. Didn't see your reply before they deleted it, unfortunately.
 
  • #5


I find this observation intriguing. It appears that there is a pattern emerging with the powers of 2 in relation to the Mersenne primes and Fibonacci numbers. This could potentially lead to further research and exploration into the underlying mathematical principles at play. It would also be interesting to see if this pattern continues for higher powers of 2 and if it can be applied to other prime numbers. This could have implications in various fields such as number theory and cryptography. Further investigation and experimentation is needed to fully understand and explain this phenomenon.
 

What is the most interesting thing you've noted?

The most interesting thing I've noted is the correlation between stress and physical health. It's fascinating how our mental and emotional state can have such a profound impact on our bodies.

How did you come across this interesting observation?

I first noticed this trend while conducting a study on the effects of stress on heart health. As I analyzed the data, I began to see a clear connection between high stress levels and increased risk of heart disease.

What implications does this have for society?

This finding has important implications for society, as it highlights the importance of addressing mental health and stress management in order to maintain overall well-being. It also emphasizes the need for more research in this area to better understand the mechanisms behind this connection.

Have other scientists found similar results?

Yes, there have been numerous studies that have found a link between stress and physical health. This reinforces the validity and significance of this observation.

How can this knowledge be applied in practical ways?

By understanding the relationship between stress and physical health, we can develop interventions and strategies to help individuals reduce their stress levels and improve their overall health. This could include promoting healthy coping mechanisms, providing access to mental health resources, and creating a supportive and stress-reducing environment in workplaces and schools.

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